Hi, my name is Bassem Saad. I'm an associate math instructor and a Ph.D. candidate, and I'm here today for about.com to introduce binomials.
What Are Binomials in Mathematics?

So a binomial is an expression made from the sum of two terms. These terms can either be variables, or real numbers, or products of both. For example, x plus two is a binomial; so is pi, times x squared, times y minus one-fifth, times y; and so is 25 times m squared, minus nine.
Binomial Examples

So here's an example of how we can build a polynomial from two binomials. We take the product of two binomials: x minus three, times x minus two. We distribute the second binomial into the first binomial, so we've got x, times x minus two, and then minus three, times x minus two. So we distribute the x into each term to give us x squared, minus two x, and we distribute the minus three into each term to give us minus three x, plus six. We combine like terms to get the final polynomial of x squared, minus five x, plus six.
Binomials

So, usually we would like to break apart a polynomial into products of binomials. For example, we could take this polynomial, which is x squared, plus four x, plus four; and we'll try to break it off into a product of a binomial of x plus a, times x plus a. In this case, we're assuming that the binomial is going to be multiplying itself. So we wish to simplify this expression to find out what a actually is. So we want to distribute the second binomial into the first, so we have x, times x plus a, plus a, times x plus a. Then we can distribute the x into x plus a to get x squared, plus ax. Then we distribute the a into x plus a to get ax, plus a squared.

Combining like terms we get x squared, plus two ax, plus a squared. Now if these expressions are equal, x squared has to equal x squared – we're ok - but four x has to equal two ax, so the coefficients have to be the same. In other words, four has to equal two a. Well we can divide two into both sides to get a equals two, but we also need four to equal a squared. Well if a equals two, you take two squared - that equals four. So we actually can break this polynomial up into a product of two binomials: x plus two, times x plus two. So now we know what binomials are.

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